Volume 2, Issue 1
Super-Resolution Inversion of Non-Stationary Seismic Traces

Jinghuai Gao, Hongling Chen, Lingling Wang & Bing Zhang

CSIAM Trans. Appl. Math., 2 (2021), pp. 131-161.

Published online: 2021-02

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  • Abstract

In reflection seismology, the inversion of subsurface reflectivity from the observed seismic traces (super-resolution inversion) plays a crucial role in target detection. Since the seismic wavelet in reflection seismic data varies with the travel time, the reflection seismic trace is non-stationary. In this case, a relative amplitude-preserving super-resolution inversion has been a challenging problem. In this paper, we propose a super-resolution inversion method for the non-stationary reflection seismic traces. We assume that the amplitude spectrum of seismic wavelet is a smooth and unimodal function, and the reflection coefficient is an arbitrary random sequence with sparsity. The proposed method can obtain not only the relative amplitude-preserving reflectivity but also the seismic wavelet. In addition, as a by-product, a special $Q$ field can be obtained.
The proposed method consists of two steps. The first step devotes to making an approximate stabilization of non-stationary seismic traces. The key points include: firstly, dividing non-stationary seismic traces into several stationary segments, then extracting wavelet amplitude spectrum from each segment and calculating $Q$ value by the wavelet amplitude spectrum between adjacent segments; secondly, using the estimated $Q$ field to compensate for the attenuation of seismic signals in sparse domain to obtain approximate stationary seismic traces. The second step is the super-resolution inversion of stationary seismic traces. The key points include: firstly, constructing the objective function, where the approximation error is measured in $L_2$ space, and adding some constraints into reflectivity and seismic wavelet to solve ill-conditioned problems; secondly, applying a Hadamard product parametrization (HPP) to transform the non-convex problem based on the $L_p (0 < p < 1)$ constraint into a series of convex optimization problems in $L_2$ space, where the convex optimization problems are solved by the singular value decomposition (SVD) method and the regularization parameters are determined by the L-curve method in the case of single-variable inversion. In this paper, the effectiveness of the proposed method is demonstrated by both synthetic data and field data.

  • AMS Subject Headings

86A22

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COPYRIGHT: © Global Science Press

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@Article{CSIAM-AM-2-131, author = {Gao , JinghuaiChen , HonglingWang , Lingling and Zhang , Bing}, title = {Super-Resolution Inversion of Non-Stationary Seismic Traces}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2021}, volume = {2}, number = {1}, pages = {131--161}, abstract = {

In reflection seismology, the inversion of subsurface reflectivity from the observed seismic traces (super-resolution inversion) plays a crucial role in target detection. Since the seismic wavelet in reflection seismic data varies with the travel time, the reflection seismic trace is non-stationary. In this case, a relative amplitude-preserving super-resolution inversion has been a challenging problem. In this paper, we propose a super-resolution inversion method for the non-stationary reflection seismic traces. We assume that the amplitude spectrum of seismic wavelet is a smooth and unimodal function, and the reflection coefficient is an arbitrary random sequence with sparsity. The proposed method can obtain not only the relative amplitude-preserving reflectivity but also the seismic wavelet. In addition, as a by-product, a special $Q$ field can be obtained.
The proposed method consists of two steps. The first step devotes to making an approximate stabilization of non-stationary seismic traces. The key points include: firstly, dividing non-stationary seismic traces into several stationary segments, then extracting wavelet amplitude spectrum from each segment and calculating $Q$ value by the wavelet amplitude spectrum between adjacent segments; secondly, using the estimated $Q$ field to compensate for the attenuation of seismic signals in sparse domain to obtain approximate stationary seismic traces. The second step is the super-resolution inversion of stationary seismic traces. The key points include: firstly, constructing the objective function, where the approximation error is measured in $L_2$ space, and adding some constraints into reflectivity and seismic wavelet to solve ill-conditioned problems; secondly, applying a Hadamard product parametrization (HPP) to transform the non-convex problem based on the $L_p (0 < p < 1)$ constraint into a series of convex optimization problems in $L_2$ space, where the convex optimization problems are solved by the singular value decomposition (SVD) method and the regularization parameters are determined by the L-curve method in the case of single-variable inversion. In this paper, the effectiveness of the proposed method is demonstrated by both synthetic data and field data.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2020-0115}, url = {http://global-sci.org/intro/article_detail/csiam-am/18657.html} }
TY - JOUR T1 - Super-Resolution Inversion of Non-Stationary Seismic Traces AU - Gao , Jinghuai AU - Chen , Hongling AU - Wang , Lingling AU - Zhang , Bing JO - CSIAM Transactions on Applied Mathematics VL - 1 SP - 131 EP - 161 PY - 2021 DA - 2021/02 SN - 2 DO - http://doi.org/10.4208/csiam-am.2020-0115 UR - https://global-sci.org/intro/article_detail/csiam-am/18657.html KW - Non-stationary seismic trace, reflectivity inversion, super-resolution, attenuation compensation, $L_p$ regularization. AB -

In reflection seismology, the inversion of subsurface reflectivity from the observed seismic traces (super-resolution inversion) plays a crucial role in target detection. Since the seismic wavelet in reflection seismic data varies with the travel time, the reflection seismic trace is non-stationary. In this case, a relative amplitude-preserving super-resolution inversion has been a challenging problem. In this paper, we propose a super-resolution inversion method for the non-stationary reflection seismic traces. We assume that the amplitude spectrum of seismic wavelet is a smooth and unimodal function, and the reflection coefficient is an arbitrary random sequence with sparsity. The proposed method can obtain not only the relative amplitude-preserving reflectivity but also the seismic wavelet. In addition, as a by-product, a special $Q$ field can be obtained.
The proposed method consists of two steps. The first step devotes to making an approximate stabilization of non-stationary seismic traces. The key points include: firstly, dividing non-stationary seismic traces into several stationary segments, then extracting wavelet amplitude spectrum from each segment and calculating $Q$ value by the wavelet amplitude spectrum between adjacent segments; secondly, using the estimated $Q$ field to compensate for the attenuation of seismic signals in sparse domain to obtain approximate stationary seismic traces. The second step is the super-resolution inversion of stationary seismic traces. The key points include: firstly, constructing the objective function, where the approximation error is measured in $L_2$ space, and adding some constraints into reflectivity and seismic wavelet to solve ill-conditioned problems; secondly, applying a Hadamard product parametrization (HPP) to transform the non-convex problem based on the $L_p (0 < p < 1)$ constraint into a series of convex optimization problems in $L_2$ space, where the convex optimization problems are solved by the singular value decomposition (SVD) method and the regularization parameters are determined by the L-curve method in the case of single-variable inversion. In this paper, the effectiveness of the proposed method is demonstrated by both synthetic data and field data.

Jinghuai Gao, Hongling Chen, Lingling Wang & Bing Zhang. (2021). Super-Resolution Inversion of Non-Stationary Seismic Traces. CSIAM Transactions on Applied Mathematics. 2 (1). 131-161. doi:10.4208/csiam-am.2020-0115
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